Fractional optimal control strategies for hepatitis B virus infection with cost-effectiveness analysis

Hepatitis B disease is a communicable disease that is caused by the hepatitis B virus and has become a significant health problem in the world. It is a contagious disease that is transmittable from person to person either horizontally or vertically. This current study is aimed at sensitivity analysis and optimal control strategies for a fractional hepatitis B epidemic model with a saturated incidence rate in the sense of the Caputo order fractional derivative approach. Fundamental properties of the proposed fractional order model are obtained and discussed. A detailed analysis of disease-free equilibrium and endemic equilibrium points is given by applying fractional calculus theory, which is a generalized version of classical calculus. Sensitivity indexes are calculated for the classical order model. Illustrative graphics that show the dependence of the sensitivity index on fractional order derivative for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document}α∈(0,1) are provided. Based on the results of the sensitivity analysis and using Pontryagin’s Maximum Principle, optimal control strategies for preventing hepatitis B infection with vaccination and treatment are considered. Fractional Euler’s method is used to carry out the numerical simulation for the proposed fractional optimal control system and the obtained results are analyzed. The results of the analysis reveal that hepatitis B disease can be prevented if necessary precautionary is taken or effective vaccination and treatment control measures are applied. The analysis of cost-effectiveness is also conducted and discussed.

Since the local asymptotical stability of the fractional order system is quite different from the ordinary integer order model, we need to define the stability of the model according to the next theorem.
Theorem 1 For α ∈ (0, 1] and z : R + × R n → R n .Consider the fractional order system of the form If all eigenvalues i (i = 1, . . ., n) of the jacobian matrix ∂f ∂z calculated at the equilibrium points of the fractional order system satisfy arg( i ) > απ 2 , then the equilibrium points are locally asymptotically stable.
Theorem 2 Suppose z : [0, b] → R is a function with n − 1 < α ≤ n, n ∈ N for all t > 0 .The following relation between Riemann-Liouville and Caputo fractional derivatives always hold and Thus, if Definition 5 ( 29 ) For z ∈ C , the one parameter and two parameters Mittag-Leffler functions with α, β ≥ 0 is defined by and satisfies the property Theorem 3 The Laplace transform of the Caputo derivative operator of order α > 0 of the function z(t) is defined as Definition 6 The Laplace transform of the Mittag-Leffler function of the form t β−1 E α,β (± t α ) is defined as Lemma 1 ( 21 ) Assume that z : [0, T] → R and 0 < α ≤ 1 .Thus, and .

Mathematical model
Based on the work of Simalene and Dlamini 15 , we extended the fractional order modeling for the hepatitis B virus infection with saturated incidence rate to fractional optimal control problem using the Caputo fractional order derivative.According to the model, the total population T(t) is divided into three sub-classes; namely susceptible, infected, and recovered individuals.The SIR model is the simplest and most common model for describing and predicting the epidemics of any contagious disease (see 30 and refs therein).Thus, the total population is given by T(t) = S(t) + I(t) + R(t).
Simalene and Dlamini 15 fractionalised hepatitis B epidemic model in the work of Khan et al. 14 presented in the Mathematical Model Formulation section using Caputo fractional order derivative.
However, we remark that the fractional order system (1) does not have appropriate dimensions.Indeed, the lefthand time dimension is (time) −α , whereas the right-hand time dimension is (time) −1 .That is, the fractional-order system (1) is only consistent when the order of differentiation α = 1 .We translated the dynamics of HBV in each population class using Caputo-fractional derivatives of order α ∈ (0, 1] .As a result, the fractional order model (FOM) described in Eq. (2) captures memory effects, which are critical for accurately defining the bio-dynamic model.As a result, the primary goal of studying fractional order derivatives is to gain a better understanding of disease dynamics.Hence, the fractional order problem characterizes the HBV spreading dynamics using the Caputo fractional derivative.
where C a D α t represents the left Caputo fractional-order derivative with derivative order α ∈ (0.1] along with initial conditions Table 1 describes the biological description, parameter's value, and their sources. In FOM (2), the quantity β α S(t)I(t) 1+γ α I(t) represents the saturated incidence rate in which the factor β α I(t) 1+γ α I(t) shows the saturation level reached whenever I(t) increases, β α I(t) measures the infection force when the disease is entering a totally susceptible population, and 1 1+γ α I(t) measures the inhibition effect from the behavioral change of susceptible individuals when their number increases.This incidence rate is more reasonable than the bilinear incidence rate since it accounts for behavioral change and prevents the contact rate from becoming unbounded by choosing suitable parameters.In this model, we also assumed that the functions S(t), I(t), R(t) and their Caputo fractional derivatives are continuous for all t ≥ 0.

Mathematical analysis
Let's denote R 3 + = {z(t) ∈ R 3 : z(t) ≥ 0} and let z(t) = (S(t), I(t), R(t)) T .Next, we investigate the non-negativity and boundedness of the FOM (2) by stating the following lemma to prove the result about the non-negativity and boundedness.
(2) From Lemma 2, we make the following remark.

Theorem 5
The closed region � = (S(t), Proof Upon adding equations of FOM (2) by taking account that S + I + R = T , we obtain Applying the Laplace transform on Eq. ( 5) using Theorem 3, we obtain Further, the simplification of Eq. ( 6) yields From Eq. (7) and using (S(0), I(0), R(0)) ∈ � , we conclude that Clearly, we obtain the boundedness of T(t) as 0 < T(t) ≤ � µ 0 α .Hence, the feasible region is positively invariant.This shows that the solution of the model is bounded.Therefore, FOM (2) is both mathematically well-posed and epidemiological meaningful.

Equilibria and stability analysis
Basically, FOM (2) has two significant equilibrium points namely: the disease-free-equilibrium (DFE) and endemic equilibrium (EE) points.Setting the right-hand side of Eq. (2) to zero in the absence of no infectives will give us the DFE of FOM (2).Let E 0 = (S(0), 0, 0) denote DFE, where S(0) = � α µ α 0 +ν α .To examine the local stability of the equilibria points, one needs to calculate the basic reproduction number by the next-generation matrix approach.Here, the matrices F and V at E 0 = � α µ α 0 +ν α , 0, 0 are calculated as where F is the non-negative matrix for new infections terms and V is the non-singular matrix for the remaining transition terms.Now, the spectral radius (ρ(FV −1 )) is calculated as ( This shows that if R 0 ≤ 1, then the disease does not spread in the population and the infection dies.On the other hand, if R 0 ≥ 1, then the disease persists in the population.The other steady state solution of FOM (2) is also known as the infective steady state (endemic equilibrium (EE)) point.Now, let E * = (S * , I * , R * ) represents EE point of FOM (2), where From Eq. ( 9), we observe that EE point of FOM (2) depends on R 0 .That means, a positive endemic equilibrium exists if R 0 > 1.
Proof The desired Jacobian matrix at E 0 becomes The eigenvalues of Eq. ( 10) are The three eigenvalues in the above equation have negative real parts such that the condition |arg( i )| = π > απ 2 as defined in Definition 4 is satisfied for i = 1, 2, 3 , Therefore, E 0 is locally asymptotically stable in if R 0 ≤ 1 .

Proof The Jacobian matrix of the FOM (2) at EE becomes
The characteristic equation of J E * from Eq. ( 11) becomes where I n E q .
( 1 2 ) , w e s e e t h a t Hence, the fractional Routh-Hurwitz con- dition is satisfied, therefore, the EE of FOM (2) is locally asymptotically stable in for α ∈ (0, 1] .

Sensitivity analysis
The sensitivity analysis of the basic reproduction number (R 0 ) plays a vital role in pointing out the most sensitive model parameters that take place in HBV transmission and control.Here, we study the relative contribution of each model parameter as it plays a significant role in disease control due to the fact that the disease incidence is related to R 0 .This leads us to determine the most sensitive parameters in the spread of the HBV infection.Before constructing an appropriate fractional order optimal control problem, we perform the sensitivity analysis of the , and , and Vol.:(0123456789) www.nature.com/scientificreports/fractional order system.We apply the normalized forward sensitivity index method to measure the importance of each parameter of FOM (2) in the disease incidence as defined in 32 .
We calculate the normalized sensitivity index of R 0 with respect to the basic parameters of the model for the integer order case (α = 1) .For example, the sensitivity index of R 0 with respect to the parameter β, � R 0 β , for the classical case (α = 1) is computed as follows In the same procedure, the sensitivity indexes of R 0 with respect to the parameters �, ν and σ are calculated and the results obtained.
The values of the sensitivity indexes of R 0 with respect to basic model parameters evaluated at DFE for FOM (2) using parameter values in Table 1 are enlisted in Table 2. Based on sensitivity indices presented in Table 2, the most sensitive parameters to disease incidence are the transmission rate (β) , vaccination rate (ν) , the disease-induced death rate (µ 1 ) , and the recovery rate (σ ) of infectious individuals.From Table 2 we observe that � R 0 ν = −0.9885which shows the parameter ν is inversely related to R 0 .This means that an increase (a decrease) of the value of ν by a small percentage will decrease (increase) R 0 by approximately the same percentage for the fractional order α = 1 .On the other hand, the parameter β is directly related to R 0 which means that an increase (a decrease) of the value of β , for instance by 10%, will increase (decrease) the value of R 0 by 10%.
However, the sensitivity index of FOM (2) parameters is determined by the derivative order α ∈ (0, 1) .The effect of the fractional derivative order α ∈ (0, 1) on model parameters ν and σ has a considerable impact on the sensitivity index of model parameters.The sensitivity indexes of ν and σ exhibit their impact on R 0 with respect to α , being very sensitive to the variation of α .The sensitivity indexes µ 0 and µ 1 , on the other hand, are substantially less sensitive to variations in the fractional-order α .The fractional order model is less sensitive than the integer order model because the model's sensitivity reduces in absolute value as the fractional derivative order α decreases.
In fact, the parameters whose index value for α = 1 is close to zero, do not vary much if we consider lower values of α .On the other hand, parameters whose index value in Table 2 is not as close to zero as the previous one vary significantly if we consider lower values of α .As a result, for the better elimination of HBV infection, it is important to increase the recovery rate (σ ) and vaccination rate (ν) .After considering the sensitivity analyses, it is determined that reasonable actions will be taken to control the transmission of the disease.

Fractional optimal control problem
In this section, we develop a fractional optimal control problem (FOCP) to prevent HBV infection.We apply the fractional optimal control methods to formulate a control mechanism for the spread of HBV infection based on sensitivity analysis.From the sensitivity analysis, it is obtained that the vaccination rate ν and the recovery rate σ have a high magnitude of sensitivity indexes.Hence our main purposes are to reduce HBV infection using vaccination (ν = u 1 ) of susceptible individuals, maximize the recovered individuals using treatment (u 2 ) of infectious individuals, and simultaneously, reduce the associated cost of the control measures.The optimality control concepts are utilized to formulate the control system.Our main objective here is to minimize the objective function J by finding a pair of optimal controls u * 1 and u * 2 such that subject to the fractional order system along the following non-negative initial conditions (ICs) In Eq. ( 13), the positive constant measures the weight constant for the infectious class I whereas B 1 and B 2 are a measure of the relative cost of the interventions associated with the vaccination and treatment controls, respectively.Here, the values 1 2 B 1 u 2 1 and 1 2 B 2 u 2 2 describe the costs associated with the vaccination and treatment www.nature.com/scientificreports/intervention strategies, respectively.It is supposed that the costs are proportional to the square of the corresponding control functions.We also assumed that the set of admissible control functions where u i is Lebesgue measurable over [0, T], u max i is the maximum value of control measures and T is a fixed final time.
The optimal control must satisfy the necessary conditions that are formulated by Pontryagin's Maximum Principle 33 .This principle converts the FOCP ( 13)-( 15) into a problem of minimizing point-wise a Hamiltonian with respect to the control u(t) = (u 1 (t), u 2 (t)) as w h e r e f (t, z(t), u(t)) is the adjoint variable of the fractional order system.
Theorem 8 Let S * , I * and R * be the optimal state solutions associated with the optimal controls u 1 , u 2 which minJ(u 1 , u 2 ) in the FOCP (13)- (15).Then, there exists a non-trivial absolutely continuous mapping where t ′ = T − t with transversality conditions Furthermore, the optimal solution (u * 1 , u * 2 ) that minimizes the fractional optimal control in O such that Proof In the fractional optimal control, the state and the control variables are positive.Thus, the necessary convexity of the objective functional in the control pair is satisfied.The admissible set O is compact.Thus, the Hamiltonian H(t, z, u, ) can be written as Using Theorem 2 the right Riemann-Liouville fractional derivative of Eq. ( 20) over [0, T] becomes where 0 ≤ t ≤ T .Equivalently, Eq. ( 21) can be written in the right Caputo-fractional derivative using Theorem 2, which yields u * 1 (t) = min max 0, , 1 , and u * 2 (t) = min max 0, www.nature.com/scientificreports/According to Lemma 1, Eq. ( 22) can be converted into the left Caputo fractional derivative as follow: For the optimal control variables u * 1 and u * 2 , we partially differentiate the Hamiltonian H using Eq. ( 21) with respect to u * i (i = 1, 2) .To this end, we solve the system ∂H ∂u * i = 0 to obtain the necessary optimality condition for the finite-dimensional optimization problem, when the control pairs u 1 and u 2 are in the interior of O as follows.
Upon simplification of Eq. ( 23), we obtain Now, taking the lower and upper bounds of Eq. ( 24) we obtain Hence, the maximality optimal controls u 1 and u 2 situations become This achieves the maximality condition

Numerical results
In this section, we utilize Pontryagin's Maximum Principle to numerically solve the FOCP (15).The fractional optimal control is obtained by solving the optimality system.We use MATLAB software with an iterative scheme used for solving the optimality system.We start by solving the state equations with a guess for the controls over the simulated time using forward fractional Euler's method.Because of the transversality conditions, the co-state equations are solved by the backward fractional Euler's method using the current iteration solutions of the state equations.Then the controls are updated by using a convex combination of the previous controls.This process is repeated and iterations stop if the values of the unknowns at the previous iterations are almost coincident with the ones of the previous iteration, that is until convergence is achieved.The solutions to the fractional control problem were performed and successfully confirmed by a classical forward-backward sweep method when α = 1 .Before performing the numerical experiments of the proposed FOM (2) and its extension to FOCP ( 13)-( 15), we will present the numerical scheme to carry out the numerical simulation.
Consider the interval of the solutions to fractional optimal control to be [0, T], which is partitioned into N equispaced grids, each of which has length h = T N .Thus, the nodes become t n = nh, (n = 1, 2, . . ., N) .Under the numerical scheme, we present the numerical solution of the nonlinear fractional differential equations (FDEs) by approximating solutions.Consider the subsequent initial value nonlinear (FDE) problem ( 22) and www.nature.com/scientificreports/For the above equation, the numerical scheme is formulated as The numerical solution of the Caputo fractional derivative of Eq. ( 14) becomes In the same procedure, the solutions to the adjoint equations become where b j,k+1 = (k − j) α − (k − j + 1) α .

Discussion
In this section, we present plots for solutions of FOM (2) for step size h = 0.01 with different fractional order α = 0.90, α = 0.95, and α = 1.00 , considering the time range is [0, 30] for simulation purposes with initial population size to be (1, 0.45, 0.05).These plots help us to verify the stability of the disease-free equilibrium and endemic equilibrium of the model.Figures 1, 2, 3, 4 and 5 demonstrate the behavior of solution curves of FOM (2) and fractional optimal control problem for different fractional order derivatives with two control measures.We first consider for the case R 0 = 0.0858 < 1 using the parameter values in Table 1.This implies that the disease will wipe out the population in the passage of time even if no intervention is applied to the system 15 .It is seen in Fig. 1a that only the susceptible population survives and in Fig. 1b the infectious individuals are going extinct.This certifies that disease-free equilibrium is locally asymptotically stable for R 0 < 1 which in turn verifies the analytical results presented in Theorem 6.It is obvious that as the fractional order α decreases in the long run, there are more susceptible individuals in the case of fractional order derivatives than in the case of integer order derivatives as depicted in Figs. 1 and 2. Next, we simulated the endemic trajectory of fractional optimal control for different fractional order values using parameter value set S = {�, β, ν, µ 0 , σ , µ 1 , γ } whose values are � = 0.03, β = 0.75, µ 0 = 0.02, σ = 0.15, ν = 0.02, µ 1 = 0.025 and γ = 0.5 in the absence of treatment and vaccination control measures, which yield R 0 = 2.8846 > 1 .This scenario implies that the disease is endemic regardless of the fractional derivative α , which indicates the disease will persist if not properly managed.Indeed, a lack of understanding regarding HBV transmission dynamics causes a delay in identifying prior phases of the disease.As a result, after knowing about HBV transmission modes, susceptible persons should take precautions or get vaccinated against HBV.This is related to the memory index of the system.Generally, lower values of the memory index correspond to increased system knowledge.Based on the memory indexes, the evolution of S(t), I(t) and R(t) of the fractional order HBV model are plotted.In Fig. 2, it is seen that the effects of fractional orders are distinctive; the solution curves for α ∈ (0, 1) show slowly in the epidemic peak and flatten faster.In particular, Fig. 2b shows the number of infectious individuals increases when the order of differentiation increases in the absence of control measures.These results reveal that if necessary interventions are not taken at a certain time, the HBV disease continues to persist in the population.So, robust strategies are required to control the transmission dynamics of the disease.
It is known that a reduction in the infectious class has a direct impact on the reduction of the HBV infection in the populations.Moreover, the infectious populations are the focus of this study.This is because infectious individuals are the population at risk of being controlled before the disease moves to the next stage of infection and progresses to cirrhosis.For this matter, we applied two control measures for the duration of one month using vaccination of susceptible individuals and treatment of infected individuals.Figure 3 demonstrates the impact of vaccination and treatment applied to the HBV infectious class and its contour profile plot for a period of at least one month with different values of the fractional order α.
In Fig. 3a, it is seen that the number of infectious individuals decreases considerably under this optimal policy for all considered values of index memory α .The combination of vaccination and treatment control measures gives the best alternative for preventing HBV infection over the specified duration of the intervention period and reduces the progression of cirrhosis effectively.Figure 3b illustrates the control profile for vaccination and treatment strategy with optimal value in the range (0, 0.6) and (0, 0.5) , respectively, for fractional deriva- tive α = 1.00 (solid) and α = 0.90 (dashed) .On the other hand, the graphics in Fig. 4 show the impact of the vaccination-only strategy applied to the fractional optimal control system and the control profile for vaccination control measure for different fractional order α .Moreover, Fig. 4a illustrates the impact of vaccination-only strategy on the dynamic behavior of HBV infection which gives a better alternative for preventing HBV and reducing the next stage of the disease.As clearly seen from the graphics, vaccination alone cannot completely control the spread of the disease but it significantly reduces the burden of HBV infection in the population which in turn reduces the progression of cirrhosis.
Figure 4b represents the control profile for the vaccination-only strategy applied to fractional optimal control in the absence of treatment control measure with variation in the order of differentiation.We observe that the vaccination control strategy is affected by the memory effect of the system.As we decrease the index of memory from α = 1.00 to α = 0.90 in the long run the administration of the vaccination control measure increases for about 15 days and then decreases afterward.In order to explore the dynamic behavior of HBV disease under the treatment control measure, we simulated the graphics for fractional optimal control in the absence of vaccination Figure 1.Evolution of (a) susceptible, (b) infected, and (c) recovered trajectories for FOM (2) with different fractional derivative order α for R 0 = 0.0858 > 1 using parameter values in Table 1.control measure as seen in Fig. 5. Figure 5a illustrates the effect of the treatment-only strategy on the infectious class which shows a positive impact in the prevention of HBV infection.Under this optimal policy, the number of infectious individuals gradually decreases as the order of fractional derivatives increases.This strategy alone cannot sufficiently minimize the burden of HBV infection, however, it gives a better alternative and most costeffective in reduce the progression of the disease into the next stage.It is seen that Fig. 5b demonstrates the control profile of the treatment-only strategy in the absence of a vaccination control measure.
In many infectious diseases, fractional derivative offers deeper insight into the system and captures the distinct memory effects of the system.In general, a lower value of fractional order derivative corresponds to less infectious individuals over the period of intervention.The number of infectious individuals is smaller in the control case for various fractional derivative orders than in the case without control.This is because the desired goal of the control measures is to reduce the number of infectious individuals and minimize the associated cost of intervention.In conclusion, the graphics clearly illustrate the effect as well as the desired goal of the controls.There is obviously a significant difference between the controlled and without-control cases.This positive influence suggests that over the intervention period, the control technique is useful in controlling the disease.

Cost-effectiveness analysis
Cost-Effectiveness Analysis (CEA) is an economic analysis of cost that helps us to compare the relative cost of two or more alternative interventions to determine and propose the most cost-effective strategy to implement with limited resources.It is important to compare the results of different control measures, with the help of calculating the Incremental Cost Effectiveness Ratio (ICER) 34 .To evaluate the cost and effectiveness of fractional optimal control for the entire intervention period, we compute the total cases averted, the total cost associated with intervention strategies, and the average cost-effectiveness ratio.To do this, we analyze the cost-effectiveness of all alternative combinations of u 1 and u 2 which is achieved: by Strategy a by implementing both controls (u 1 , u 2 = 0) , Strategy b by implementing only vaccination (u 1 = 0 and u 2 = 0) , and Strategy c by implementing only treatment (u 1 = 0 and u 2 � = 0) .This ratio is used to compare the differences between the costs and health benefits of two alternative intervention strategies that compete for the same resources and is defined as Following 35 the function F : R + → [0, 1] known as the efficacy function is used to measure the proportional differences in the number of infected individuals after the application of the treatment and vaccination compartment by comparing the number of infectious individuals at t with its initial value I(0) is defined as (27)

ICER =
Difference between cost benefits Difference between health benefits   In Eq. ( 28), the curve I * (t) is the optimal solution associated with the fractional optimal control and I(0) is the initial condition.We observe from Fig. 6 that the efficacy function exhibits the inverse tendency of infected individuals and is the highest when F(t) is unity.The total cases averted (AV) by the intervention during the time period T is given by In Eq. ( 29), the trajectory I * (t) is the optimal solution associated with the fractional optimal control.The quantity I(0) represents the corresponding initial condition where this initial condition is obtained as the equilibrium proportion Ĩ(t) of FOM (2) with no post-exposure intervention does not depend on time.Thus, T × I(0) = T 0 Ĩ(t)dt represents the total infectious cases over a period of T days.
Effectiveness is the proportion of cases averted on the total cases possible under no intervention and given as 32,35 .Eq. ( 30) is used to compare different epidemiological scenarios, in which we choose dimensionless measures for effectiveness.The total cost (TC) associated with the intervention is given by In Eq. ( 31), the constants C 1 and C 2 correspond to the per person unit cost of vaccination of susceptible popula- tion and treatment of infected population, respectively.Following 35 , we define the average cost-effectiveness ratio (ACER) as Eq. ( 32) deals with a single intervention strategy and evaluates that intervention against no intervention or current practices.Tables 3, 4 and 5 summarize the cost-effectiveness measures of fractional optimal control with different combinations of controls.
Based on the model simulation results and using Tables 3, 4 and 5, we rank the strategies in order of increasing effectiveness for the classical order α = 1.
The comparison between strategies b and c shows that ICER(c) < ICER(b).The lower ICER value corresponds to strategy b strongly dominating strategy c, which is more costly and less effective.Therefore, strategy b is excluded from the set of alternative interventions so that it does not consume limited resources.We arrange the remaining strategies by increasing effectiveness and recalculating the ICER for strategies a and c.Again, the comparison between strategies a and c show that ICER(c) < ICER(a).This implies that strategy a strongly dominated strategy c.It is more costly and consumes limited resources, therefore, we exclude strategy a from the list of intervention strategies.With this result, we conclude that strategy c (treatment of infectious individuals) has the least ICER.Therefore, strategy c is more cost-effective than strategy a and b in the case of the classical order model as indicated in Table 4.
The ICER was calculated in the same way for the fractional order derivative situations.Table 5   www.nature.com/scientificreports/effectiveness ratio drops as the derivative order decreases, we exclude the intervention with the highest total cost first.As a result, α = 1.00 and α = 0.95 are excluded.Finally, strategy c (treatment of infectious individuals) with fractional derivative order α = 0.90 is the most cost-effective intervention.This conclusion, however, should be interpreted with caution due to ambiguities surrounding the parameter values, but it provides crucial deeper insight into the prevention of HBV disease with variation in the order of differentiation.

Conclusion
In this paper, we derived and analyzed a deterministic fractional order model for the transmission of hepatitis B disease that includes treatment and vaccination control measures with a saturated incidence rate.It is known that treatment and vaccination of hepatitis B disease reduce the risk of progression and so it is desirable to apply control measures in these efforts to prevent the disease.The proposed model consists of susceptible (S), infected (I), and recovered (R) individuals, which are considered to be the most basic components of HBV infection.The positivity of solutions and invariant region of the fractional order model are discussed to show the biological significance of the system.The equilibrium points for disease-free steady state and infected steady state are computed, and an investigation of their local stability is performed.Thus, the condition under which the fractional order system's disease-free equilibrium points are stable is established.The basic reproduction number is calculated to be R 0 = 0.0858 < 1 and provides important information about the dynamics of the disease in the future.The sensitivity analysis of the integer order model and fractional order model for α ∈ (0, 1) cases are computed.The comparison between the sensitivity analyses of the classical model and the fractional order model shows that the sensitivity analysis of the fractional order model depends on the order of fractional derivatives whereas the classical models do not.
We investigated the fractional optimal control problem by the application of the optimal control theory and used Pontryagin's Minimum Principle to provide the necessary conditions needed for the existence of its optimal solution.Optimal control interventions involving vaccination of susceptible populations and treatment of infective individuals are incorporated into the fractional order model, The fractional optimal control problem was analyzed theoretically and numerically.The numerical simulation showed that with the help of vaccination and treatment controls over a specified period of time we can eliminate HBV infection.

Figure 2 .
Figure 2. Dynamic behavior of (a) susceptible class, (b) infected class and (c) recovered class for FOM (2) with different fractional α for R 0 = 2.8846 > 1 using parameter value set S.

Figure 3 .
Figure 3. (a) Impact of the optimal combination of vaccination and treatment control measures on infectious class and (b) the control profile for vaccination u 1 and treatment u 2 strategy using parameter value set S with different fractional derivative order α.

Figure 4 .
Figure 4. (a) Impact of the optimal use of vaccination only control strategy on infectious class and (b) the control profile for vaccination u 1 strategy using parameter value set S with different fractional derivative order α.

Figure 5 .
Figure 5. (a) Impact of the optimal use of treatment only strategy on infectious class and (b) the control profile for treatment u 2 strategy using parameter value set S with different fractional derivative order.

Figure 6 .
Figure 6.Evolution of the efficacy function of the fractional optimal control problem with different values of the fractional order α.

Table 3 .
Summary of cost-effectiveness measures of hepatitis B disease with optimal use of vaccination and treatment control measures.

Table 4 .
Incremental cost-effectiveness ratio of hepatitis B disease in the case of classical order α = 1.00.

Table 5 .
Incremental cost-effectiveness ratio for strategy c with variation in the order of differentiation.